Depth of factors of square free monomial ideals

TL;DR

This paper establishes conditions under which the depth of square free monomial ideals and their quotients equals a specific degree, confirming Stanley's Conjecture in these cases.

Contribution

It provides new criteria for the depth of square free monomial ideals and quotients, verifying Stanley's Conjecture under these conditions.

Findings

Depth equals degree d under certain combinatorial conditions.

Stanley's Conjecture holds for these classes of ideals.

Provides explicit bounds involving the number of monomials.

Abstract

Let $I$ be an ideal of a polynomial algebra over a field, generated by $r$ square free monomials of degree $d$. If $r$ is bigger (or equal, if $I$ is not principal) than the number of square free monomials of $I$ of degree $d+1$, then $\depth_SI= d$. Let $J\subsetneq I$, $J\not =0$ be generated by square free monomials of degree $\geq d+1$. If $r$ is bigger than the number of square free monomials of $I\setminus J$ of degree $d+1$, or more generally the Stanley depth of $I/J$ is $d$, then $\depth_SI/J= d$. In particular, Stanley's Conjecture holds in theses cases.